Finite Group Databasebeta

← #10690954

Subgroups — #10690954

Order 768.

902 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82834C211yesmaximal, characteristic
2C2×#4664C211yesmaximal, characteristic
3C2×C3×#328C211yesmaximal, characteristic, complemented
4C2×#4538C211yesmaximal, characteristic, complemented
5#69955C211yesmaximal, characteristic
6#67675C211yesmaximal, characteristic, complemented
7#67612C211yesmaximal, characteristic, complemented
8#9154—31nomaximal, Sylow
9C23×C4×D3C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C2×#946C412yescomplemented
12C22×(C2×D3)⋊C4—21no—
13C2×C3×C22⋊C8C2211yescharacteristic
14C2×C3×#95C2211yescharacteristic
15#5212C2211yescharacteristic
16C2×C3⋊#95—21no—
17C2×C3⋊#95C2211yescharacteristic
18C2×#830C2211yescharacteristic
19C2×(C2×D3)⋊C8—22no—
20#4989—22no—
21#4987C412yescomplemented
22#4955—21no—
23#4946—21no—
24#4664—22no—
25#4664—22no—
26C3×#328—22no—
27#4538—22no—
28#2848—31no—
29C2×#381—31no—
30#1779—31no—
31C2×#328—31no—
32C2×#328D311yescharacteristic, complemented
33#1294—31no—
34#1294—31no—
35C24×D3D411yescharacteristic
36C22×C3×Q8C2×C411yescharacteristic
37C23×C3×C4—21no—
38C23×C3×C4C2311yescharacteristic
39C23×Dic3—21no—
40C23×Dic3D411yescharacteristic
41C22×C4×D3—21no—
42C22×C4×D3—21no—
43C22×C4×D3—21no—
44C22×C4×D3—21no—
45C22×C4×D3—44no—
46C22×C4×D3C2×C412yes—
47C22×C4×D3D412yes—
48C22×C3×C8—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×C22⋊Q8—21no—
52C3×C22⋊Q8—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4C2×C411yescharacteristic
55C2×C3×C22⋊C4—21no—
56C2×#217—21no—
57#946—44no—
58C2×(C2×D3)⋊C4—24no—
59C2×(C2×D3)⋊C4—22no—
60C2×(C2×D3)⋊C4—22no—
61C2×(C2×D3)⋊C4—44no—
62C2×C4⋊Dic3C2×C412yes—
63C2×Dic3⋊C4—22no—
64C22×C3⋊C8—21no—
65C3×C22⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×#95—21no—
68C3×#95D412yes—
69C3⋊#95—24no—
70C3⋊#95—21no—
71C3⋊#95—21no—
72C3⋊#95—22no—
73C3⋊#95D412yes—
74#830—21no—
75#830—21no—
76(C2×D3)⋊C8—44no—
77C24×C4—31no—
78C2×C22⋊Q8—32no—
79C2×C22⋊Q8C2×D311yescharacteristic
80C22×C22⋊C4—61no—
81C2×C22⋊C8—31no—
82C2×C22⋊C8—62no—
83C2×C22⋊C8C2×D311yescharacteristic
84#385—61no—
85#385—62no—
86#381—62no—
87#381—62no—
88#380—31no—
89#380—32no—
90#379—61no—
91C2×#95—31no—
92C2×#95—61no—
93C2×#95C2×D311yescharacteristic
94#328—22no—
95#328—62no—
96C24×C3C2×D411yescharacteristic
97C23×D3—21no—
98C23×D3—21no—
99C23×D3—24no—
100C23×D3—21no—
101C23×D3—21no—
102C23×D3—21no—
103C23×D3—21no—
104C23×D3—42no—
105C23×D3C22⋊C412yes—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—42no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—22no—
112C22×C3×C4—42no—
113C22×C3×C4—42no—
114C22×C3×C4—42no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4C22×C411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4#4111yescharacteristic
122C22×C3×C4#4111yescharacteristic
123C22×Dic3—21no—
124C22×Dic3—21no—
125C22×Dic3—22no—
126C22×Dic3—22no—
127C22×Dic3—42no—
128C22×Dic3—42no—
129C22×Dic3—42no—
130C22×Dic3—41no—
131C22×Dic3—41no—
132C22×Dic3C22⋊C412yes—
133C2×C4×D3—24no—
134C2×C4×D3—24no—
135C2×C4×D3—22no—
136C2×C4×D3—22no—
137C2×C4×D3—22no—
138C2×C4×D3—22no—
139C2×C4×D3—42no—
140C2×C4×D3—42no—
141C2×C4×D3—44no—
142C2×C4×D3—44no—
143C2×C4×D3—44no—
144C2×C4×D3—44no—
145C2×C3×C8—42no—
146C2×C3×C8—41no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—41no—
150C3×C4⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153#217—41no—
154#217—41no—
155(C2×D3)⋊C4—44no—
156(C2×D3)⋊C4—48no—
157(C2×D3)⋊C4—44no—
158C4⋊Dic3—42no—
159Dic3⋊C4—44no—
160C2×C3⋊C8—42no—
161C2×C3⋊C8—41no—
162C25—31no—
163C22×Q8C4×D311yescharacteristic
164C23×C4—21no—
165C23×C4—31no—
166C23×C4—32no—
167C23×C4—32no—
168C23×C4—61no—
169C23×C4—61no—
170C23×C4—61no—
171C23×C4—61no—
172C23×C4—61no—
173C23×C4—124no—
174C23×C4C22×D311yescharacteristic
175C22×C8—21no—
176C22×C8—61no—
177C22⋊Q8—21no—
178C22⋊Q8—21no—
179C22⋊Q8—21no—
180C22⋊Q8—21no—
181C22⋊Q8—124no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—32no—
184C2×C4⋊C4—62no—
185C2×C4⋊C4C4×D311yescharacteristic
186C2×C22⋊C4—21no—
187C2×C22⋊C4—62no—
188C2×C22⋊C4—62no—
189C2×C22⋊C4—64no—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—124no—
192C22⋊C8—21no—
193C22⋊C8—21no—
194C22⋊C8—61no—
195C22⋊C8—61no—
196C22⋊C8—124no—
197#95—21no—
198#95—32no—
199#95—62no—
200#95—64no—
201#95—61no—
202#95—61no—
203#95#6712yes—
204C23×C3—22no—
205C23×C3—22no—
206C23×C3C2×C22⋊C411yescharacteristic
207C23×C3#12011yescharacteristic
208C23×C3#12311yescharacteristic
209C23×C3#13511yescharacteristic
210C23×C3#13511yescharacteristic
211C23×C3#13711yescharacteristic
212C23×C3#13711yescharacteristic
213C22×D3—24no—
214C22×D3—24no—
215C22×D3—22no—
216C22×D3—22no—
217C22×D3—22no—
218C22×D3—22no—
219C22×D3—22no—
220C22×D3—22no—
221C22×D3—44no—
222C22×D3—44no—
223C22×D3—44no—
224C22×D3—44no—
225C22×D3—42no—
226C22×D3—42no—
227C22×D3—41no—
228C22×D3—41no—
229C22×D3—41no—
230C22×D3—41no—
231C22×D3#9914yes—
232C3×Q8—81no—
233C3×Q8—81no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—82no—
248C2×C3×C4—82no—
249C2×C3×C4C4×D411yescharacteristic
250C2×C3×C4C4×D411yescharacteristic
251C2×C3×C4#12111yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×Dic3—22no—
254C2×Dic3—22no—
255C2×Dic3—22no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264C2×Dic3—82no—
265C2×Dic3—82no—
266C4×D3—44no—
267C4×D3—44no—
268C4×D3—44no—
269C4×D3—44no—
270C3×C8—81no—
271C3⋊C8—81no—
272C24—32no—
273C24—64no—
274C24—61no—
275C24—61no—
276C24—61no—
277C24—61no—
278C24—61no—
279C24—61no—
280C24—122no—
281C24D3×D411yescharacteristic
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—42no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—32no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—64no—
295C22×C4—64no—
296C22×C4—61no—
297C22×C4—61no—
298C22×C4—62no—
299C22×C4—62no—
300C22×C4—62no—
301C22×C4—62no—
302C22×C4—62no—
303C22×C4—62no—
304C22×C4—122no—
305C22×C4—122no—
306C22×C4—124no—
307C22×C4—124no—
308C22×C4—124no—
309C22×C4—124no—
310C22×C4—122no—
311C22×C4—122no—
312C22×C4—122no—
313C22×C4—121no—
314C22×C4—121no—
315C22×C4C2×C4×D311yescharacteristic
316C22×C4#23511yescharacteristic
317C22×C4D3×D411yescharacteristic
318C22×C4D3×D411yescharacteristic
319C22×C4#23711yescharacteristic
320C22×C4C2×#6711yescharacteristic
321C2×C8—42no—
322C2×C8—41no—
323C2×C8—122no—
324C2×C8—121no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—122no—
330C4⋊C4—124no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—124no—
334C22⋊C4—128no—
335C22⋊C4—124no—
336C22⋊C4—121no—
337C22⋊C4—121no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—22no—
343C22×C3—22no—
344C22×C3—22no—
345C22×C3—22no—
346C22×C3#38611yescharacteristic
347C22×C3C2×#9912yes—
348C22×C3#41611yescharacteristic
349C22×C3#41911yescharacteristic
350C22×C3#42112yes—
351C22×C3#45211yescharacteristic
352C22×C3#45211yescharacteristic
353C22×C3#48511yescharacteristic
354C22×C3#48511yescharacteristic
355C2×D3—24no—
356C2×D3—24no—
357C2×D3—24no—
358C2×D3—44no—
359C2×D3—44no—
360C2×D3—44no—
361C2×D3—44no—
362C2×D3—42no—
363C2×D3—42no—
364C2×D3—42no—
365C2×D3—42no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—82no—
371C3×C4—81no—
372C3×C4—81no—
373Dic3—42no—
374Dic3—82no—
375Dic3—82no—
376C23—22no—
377C23—22no—
378C23—34no—
379C23—64no—
380C23—64no—
381C23—62no—
382C23—62no—
383C23—62no—
384C23—62no—
385C23—62no—
386C23—62no—
387C23—124no—
388C23—124no—
389C23—124no—
390C23—124no—
391C23—122no—
392C23—122no—
393C23—121no—
394C23—121no—
395C23—121no—
396C23—121no—
397C23D3×C22⋊C411yescharacteristic
398C23#88311yescharacteristic
399C23#91511yescharacteristic
400C23#91611yescharacteristic
401C23#91811yescharacteristic
402C23#91911yescharacteristic
403C23#93711yescharacteristic
404Q8—81no—
405Q8—81no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—62no—
420C2×C4—62no—
421C2×C4—62no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—124no—
425C2×C4—124no—
426C2×C4—124no—
427C2×C4—124no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4—122no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4—122no—
435C2×C4—122no—
436C2×C4—242no—
437C2×C4—242no—
438C2×C4#67⋊C411yescharacteristic
439C2×C4#88411yescharacteristic
440C2×C4C4×#6711yescharacteristic
441C2×C4#93811yescharacteristic
442C8—81no—
443C8—241no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3#151112yes—
449C2×C3#151612yes—
450C2×C3#187311yescharacteristic
451C2×C3#189711yescharacteristic
452C2×C3#189811yescharacteristic
453D3—44no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—64no—
463C22—64no—
464C22—64no—
465C22—124no—
466C22—124no—
467C22—124no—
468C22—124no—
469C22—122no—
470C22—122no—
471C22—122no—
472C22—122no—
473C22D3×#9912yes—
474C22#480211yescharacteristic
475C22#480411yescharacteristic
476C22#481411yescharacteristic
477C22#481511yescharacteristic
478C22#482112yes—
479C22#495611yescharacteristic
480C22#517311yescharacteristic
481C22#518711yescharacteristic
482C4—21no—
483C4—21no—
484C4—21no—
485C4—21no—
486C4—82no—
487C4—81no—
488C4—81no—
489C4—122no—
490C4—242no—
491C4—242no—
492C3#915411yescharacteristic, complemented, Sylow
493C2—21no—
494C2—21no—
495C2—21no—
496C2—21no—
497C2—124no—
498C2#6833612yes—
499C2#6833812yes—
500C2#7003611yescharacteristic
501C2#7026911yescharacteristic
502C2#7038311yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#328C2
socleC23×C3C2×C22⋊C4
prime radicalC24×D3D4
cosocleC23×C3×C4C23
hypercenterC2×#328D3
hypocenterC3#9154
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×C3×#328C2semidirect
4. C2×#4538C2semidirect
6. #67675C2semidirect
7. #67612C2semidirect
11. C2×#946C4semidirect
21. #4987C4semidirect
32. C2×#328D3semidirect
492. C3#9154semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#828342
C2×#46642
C2×C3×#3282
C2×#45382
#699552
#676752
#676122
#91543